For a seven-day period, you shall bring a fire offering to the Lord . On the eighth day, it shall be a holy occasion for you, and you shall bring a fire offering to the Lord . It is a [day of] detention. You shall not perform any work of labor.

— Leviticus 23:39

When it began:

The festival of Shemini Atzeret was given to the Jewish people during their 40-year sojourn in the desert. It directly follows the seven days of Sukkot, and therefore Shemini Atzeret is literally the eighth day; but it is an independent one-day holiday. God says to Israel, "I have detained you to remain with Me." This is analogous to a king who invited his children to feast with him for a certain number of days, and when the time came for them to leave, he said: "My children! Please, stay with me just one more day; it is difficult for me to part with you!"

How it’s observed today:

Shemini Atzeret is celebrated as a full festival, with no manual labor permitted (like Shabbat) and with special prayer services and festive meals. The special prayer Geshem (Prayer for Rain) is recited during the Mussaf service – because this heralds the coming winter season where the rains are needed for the new crops.

Customs:

Yizkor, the prayer for remembrance of deceased parents/ancestors is recited.